A Proposed Method for Finding Stress and Allowable Pressure in Cylinders Cylinder s with Radial Nozzles
Les M. Bildy Codeware
ABSTRACT A simple technique is presented for calculating the local primary stress from internal pressure in nozzle openings. Nozzle internal projections, reinforcing pads and fillet welds are considered for nozzles on cylinders. The technique uses beam on elastic foundation theory and extends the work of W. W. L. McBride and W. S. Jacobs [7]. A study comparing the proposed method with ASME Section VIII, Division 1 [2] rules and finite element analysis (FEA) is presented for the range of geometries listed in WRC-Bulletin 335. The proposed method predicts
by T. P Pastor and J. Hechmer [8], and that the results be properly interpreted. Because of the uncertainties involved in the application of FEA in determining primary stress, WRC-429 tends to discourage the general use of FEA for this purpose. The Code provides no direct method for calculating the magnitude of the primary stress at a nozzle-to-cylinder intersection. The objective of this paper is to propose a simple method to calculate the maximum primary stress produced by internal pressure at nozzle-to-cylinder intersections.
BACKGROUND
and cylinder. The McBride and Jacobs method includes consideration of a bending moment that is assumed to be produced by the way the nozzle geometry intersects with the cylinder. The limit assumed to resist this pressure induced nozzle bending is taken to be [95/(F y)1/2]T, rounded to 16T for steel. A modified version of this method is the basis for the VIII-1, Appendix 1-7(b) rules for large openings in cylindrical shells. McBride and Jacobs’ work does not specifically address the more common case of smaller nozzles for which the proposed procedure has been developed.
DISCUSSION OF METHOD The proposed method is based on the assumption of elastic equilibrium. A pressure area calculation similar to the technique used when determining tensile stress when applying VIII-1, Appendix 1-7(b) is employed. It differs from the current Code rules and from McBride and Jacobs in the following ways: (a) The area in the cylinder near the opening that is considered to reinforce the opening is taken to be a simple function of the shell thickness, namely, 8T for integrally reinforced nozzles, 8(T + t e) for nozzles with “wide” pads, and 10T for all other pad reinforced nozzles. 8T is approximately the tensile half of a noncompact beam having a limiting width-thickness ratio of [95/(F y)1/2]T (Manual of Steel Construction [5]). It is not a function of the beam on elastic foundation limit (RT) 1/2 due to the nature of the nozzle discontinuity in the cylinder. This is because a “small” nozzle does not present an axisymmetric discontinuity to the cylinder; hence, (RT) 1/2 is not an indication of the attenuation distance in the cylinder. However, (R ntn)1/2 does describe the attenuation distance in the nozzle because the cylinder does restrain the nozzle in an approximately axisymmetric manner. (b) The maximum local primary membrane stress (PL) is determined from the calculated average membrane stress using a linear stress distribution. The assumed stress distribution is identical to that produced by a beam subject to a point load at the extreme fiber (see Figure
Figure 2 Assumed Stress Distribution at Shell Discontinuity DESIGN METHOD Areas contributing to nozzle reinforcement are described in Figure 3:
Figure 3 Areas Contributing to Reinforcement
A41 = 0.5Leg412
(9)
2 42
A42 = 0.5Leg
(10)
A43 = 0.5Leg432
(11)
A5: Area contributed by pad A5 = wte,
(12) w <= L R
f N: force in nozzle outside of vessel f N = PRn(LH – T)
(13)
f S: force in shell f S = PR(LR+tn)
(14)
f Y: force due to discontinuity (finished opening in the shell) f Y = PRRn
(15)
Save: local average primary membrane stress Save = (f N + f S + f Y) / (A1 + A2 + A3 + A41 + A42 + A43 + A5)
(16)
PL: local maximum primary membrane stress PL = 2Save – PR/T
(17)
Note: Replace T with T + te for nozzles with wide pads where w>=8(T+te)
Derivation of Nozzle Limit Pressure AT: Total tensile area near opening AT = A1 + A2 + A3 + A41 + A42 + A43+ A5
FT: Total tensile force near opening FT = f N + f S + f Y
(18)
mate stress (Su) and vessel burst pressure (P) were taken from WRC335. An entry of “burst” in column S u indicates that the pressure listed in column P was taken from a physical burst test. A numeric entry in column Su indicates that S u was used in WRC-335 to calculate the theoretical vessel burst pressure presented here. ASME VIII-1 area replacement calculations are listed in the A /A a r column where A a is the available area of reinforcement and A r is the area of reinforcement required. For comparison purposes the allowable stress was taken to be S u. The ratio Aa /Ar can be taken as an indicator of the accuracy of the VIII-1 calculation. Ideally, the ratio should be 1.0; however, deviation from the ideal ratio is not linear. This is due to the way the VIII-1 shell contributing area A 1 is calculated. VIII-1 A1 is a function of geometry and pressure. Lowering pressure both decreases area required and increases area available. In WRC-335 and Tables 1 and 2 this has the net effect of making the Code rules appear more conservative. The relationship between nozzle stress and nozzle internal projection was not investigated by either WRC-335 or McBride and Jacobs. Since the proposed method includes consideration of this detail, several FEA comparisons that include a nozzle internal projection were performed.
DISCUSSION OF RESULTS A review of the results in Table 1 reveals that the proposed method agrees well with both FEA and burst test data from WRC-335 with two notable exceptions. In Table 1 where D o = 24" the FEA model and WRC-335 indicate a lower stress than the proposed method. A closer examination of the FEA models gives the following possible explanation. In these cases, the region of high stress is not confined to the area near the shell longitudinal axis. One explanation of this is that when the geometry is more flexible, strain redistribution around the circumference of the nozzle occurs. A similar stress distribution was also observed in the FEA models for cases where the R n /R ratio was larger than about 0.5. The conclusion could be made that the assumption of uniaxial strain breaks down for thin wall vessels having D/T > 200 and for
brane stress for the case of “thick” nozzles or thick reinforcing pads generally seems to occur about 45 degrees from the shell longitudinal axis. Despite this, the magnitude of the stress P L per the proposed method agrees reasonable well with the FEA results. The first entry in Table 2 provides an explanation of the unusual failure of the pad reinforced test model noted in WRC-335 3.1.6. E. C. Rodabaugh [6] indicated that the failure location was along the longitudinal axis (WRC-335 location B failure), whereas all other pad reinforced nozzles failed at location A. The proposed method and FEA agree in predicting that failure could occur at location B for this case. Both the proposed method and FEA also agree in predicting that failure would likely not occur at location B for any of the other pad reinforced WRC-335 models investigated.
CONCLUSIONS The present paper provides several insights into the following VIII1 design rules for nozzles in cylinders: Paragraph UG-37 The rules in UG-37 concerning A 1 are based on the nozzle diameter. The present study shows that a more accurate analysis is obtained if the area contributed by the shell is taken to be a function of the shell thickness only. This means that in general the smaller the nozzle the more conservative the Code becomes. This trend indicates that the exemptions found in UG-36(c)(3)(a) are consistent in principle with the findings in this paper. The data in this paper support the conclusion that even with the new higher allowable stresses the Code requires additional reinforcement for many moderately sized nozzles where a more detailed stress analysis would indicate otherwise. In addition to increasing costs, adding more material than required may be detrimental to the fatigue life of the vessel. This effect is more pronounced if the additional reinforcement is provided by a pad. Paragraph UG-42 Another interesting conclusion concerns the Code provisions for overlapping limits of reinforcement. If the true limit of reinforcement in
of such a problem is found in McBride and Jacobs Case 1. Table 2 shows that the 169.24 inch diameter opening meets Appendix 1-7(a) and UG-37 rules for a pressure of 114 psi. Without the provisions of Appendix 1-7(b), local primary membrane stresses in excess of yield (38 ksi for SA-516 70) would be present in the shell.
Appendix 1-7(b) The proposed method does not consider pressure induced bending moments. In spite of this, the results are in good agreement with the strain gauge measurements reported by McBride and Jacobs. It is the opinion of the author that the pressure induced bending moment is resisted by the attached shell along the outer perimeter of the nozzle. It does not appear that the pressure induced bending moment affects the primary stress analysis of the nozzle at the longitudinal axis. Nomenclature Aa Available area of reinforcement by VIII-1 rules, in 2 Ar Required area of reinforcement by VIII-rules, in 2 Fy Material yield stress, ksi LH Stress attenuation distance in nozzle, in LR Stress attenuation distance in shell, in Leg41 Outer nozzle to shell weld, in Leg42 Pad to shell weld, in Leg43 Inner nozzle to shell weld, in h Nozzle inside projection, in PL Shell maximum local primary membrane stress, psi R Shell inner radius, in Rn Nozzle inner radius, in SV Shell allowable stress, psi (VIII-1, UG-37) SN Nozzle allowable stress, psi (VIII-1, UG-37) SP Reinforcing pad allowable stress, psi (VIII-1, UG-37) SY Yield stress, psi S Lesser of shell, nozzle, or pad allowable stress, psi (II-D) T Shell thickness, in tn Nozzle thickness, in
Ref
DO
T
dO
t
P
Su
PL
PL
Ratio
Aa /Ar
No
in
in
in
in
psi
ksi
FEA
proposed
(5)
12.75
0.375
6.625
0.28
4,100
burst
147.4
148.7
1.009
0.066
(g)
12.75
0.375
6.625
0.28
4,100
65.6
133.7
124.2
0.929
0.251
(5)(c)
12.75
0.375
7.003
0.469
4,100
burst
109.7
117.8
1.074
0.218
(g)
12.75
0.375
7.003
0.469
4,100
65.6
89.7
83.5
0.931
0.58
(a)
(5)(c)
12.75
0.375
7.445
0.690
4,100
burst
95.5(d)
88.3
0.935
0.36
(g)
12.75
0.375
7.445
0.690
4,100
65.6
56.2(d)
53.5
0.952
0.889
(5)
18.00
0.375
3.00
0.188
2,330
burst
77.8
85.5
1.099
(b)
(5)
18.00
0.375
4.00
0.188
2,330
burst
99.0
100.5
1.015
0.116
(5)
18.00
0.375
4.00
0.250
2,330
burst
89.6
92.6
1.033
0.24
(5)
18.00
0.375
4.00
0.375
2,330
burst
73.3
73.0
0.996
0.584
(5)(c)
18.00
0.375
6.00
0.250
2,330
burst
113.2
120.4
1.064
0.127(c)
(5)(c)
18.00
0.375
6.00
0.375
2,330
burst
87.0
95.5
1.098
0.325(c)
(5)
8.625
0.322
8.625
0.322
4,600
burst
132.3
147.3
1.113
0.042(e)
(5)
8.625
0.322
4.50
0.237
4,600
burst
111.7
122.0
1.092
0.118
(5)
8.625
0.500
8.625
0.500
7,220
58.7
111.5
121.4
1.089
0.041(e)
(5)(c) (5)
12.75 24.00
0.687 0.312
6.625 4.50
0.432 0.237
6,760 1,970
59.4 79.9
105.7 129.8
113.5 141.2
1.074 1.088
0.082(c) 0.419
(g)
24.00
0.312
4.50
0.237
1,970
79.9
109.5
108.2
0.988
0.520
24.00
0.312
12.75
0.250
1,580
84.3
194.4
241.3
1.241
0.269
24.00
0.312
12.75
0.250
1,580
84.3
161.8
182.7
1.129
0.450
(5)
24.00
0.312
24.0
0.312
1,620
84.3
247.8
285.1
1.151
0.208(e)(c)
(5)
5.983
0.193
3.59
0.116
3,250
62.4
132.6
130.3
0.983
(5)
24.00
0.104
12.75
0.102
225
49.0
123.5
205.8
1.666
0.60
(5)
12.50
0.281
10.5
0.250
2,200
60.9
150.4
151.6
1.008
0.028(c)
(5)
6.50
0.176
4.50
0.144
2,920
61.1
136.2
150.9
1.108
0.075(c)
(5)(f)
4.50
0.237
1.32
0.133
6,350
60.3
78.1
79.7
1.020
(b)
(5)
4.50
0.237
2.38
0.154
6,175
60.4
95.6
105.7
1.106
(b)
(5)
4.50
0.237
3.50
0.216
6,100
60.3
92.8
102.8
1.108
0.148
(5) (g)
(b)
Ref. No.
D O in
T in
d O in
t in
W in
t e in
P psi
S ksi
(5)(b) (5) (5) (5) (6) (6) (6) (6) (h) (6) (6) (h) (6) (h)
12.75 8.625 36.000 36.000 169.24 169.24 169.24 169.24 169.24 79.125 79.125 79.125 158.811 158.811 61.50 61.50 61.50
0.375 0.500 0.375 0.675 0.620 0.620 0.620 0.620 0.620 0.563 0.563 0.563 1.406 1.406 0.75 0.75 0.75
6.625 8.625 12.75 12.75 60.00 60.00 60.00 60.00 60.00 30.75 30.75 30.75 95.354 95.354 12.75 12.75 12.75
0.28 0.500 0.500 0.500 0.470 0.470 0.470 0.470 0.470 0.563 0.563 0.563 2.677 2.677 0.688 0.688 0.688
2.15 4.313 18.00 18.00 17.24 17.24 17.24 17.24 17.24 8.63 8.63 8.63 24.76 24.76 2.50 5.00 2.00
0.125 0.500 0.375 0.675 0.620 0.620 0.620 0.620 0.620 0.563 0.563 0.563 2.087 2.087 0.375 0.25 0.75
4,440 8,750 1,810 3,115 28 57 85 114 114 135 273 273 457 457 350 350 350
73.2 76.1 85.8 85.8 12.9(e) 25.2(e) 37.1(e) 48.4(g) 18.2(g) 42.7(g) 43.9(g)
PL FEA ksi 127.7(d) 68.3 98.4 86.9 11.0 22.5 33.5 44.9 41.5 18.6 39.3 33.6 39.0 33.5 20.3 21.4 17.8(i)
PL proposed ksi 124.1 80.2 110.6 99.4 11.5 23.4 34.9 46.8 43.1 18.0 36.3 29.9 33.7 30.1 20.6 20.6 16.1
Ratio
0.972 1.174 1.124 1.144 1.046 1.040 1.042 1.042 1.034 0.968 0.924 0.890 0.864 0.899 1.015 0.963 0.904
Table 2: Comparison of Design Methods for Pad Reinforced Nozzle to Cylinder Connections
Notes: (a) VIII-1 calculation performed using Sv = Sn = S p = 20 ksi for SA-516 70.
Aa/Ar VIII-1 (a) 0.31(f) 0.68(c)(f) 1.10(f) 0.72(f) 7.40 3.21 1.82 1.09 1.14 2.53 0.75 0.85 0.71 0.78 0.95 1.03 1.25
Appendix A - September 1, 2000
Subsequent to publication, further verification of the method outlined in the paper has been performed. The purpose of this Appendix is to present a comparison between linear elastic FEA and the proposed method over a wider range of geometries. Calculations (in the form of area required divided by area available) using the ASME Code, A99 Addenda of Division 1 and Division 2 are also presented for comparison. Note that "duplicate" D/t ratios in the graphs represent different shell and nozzle diameter/thickness combinations. As a result of this investigation it seemed desirable to add two additional geometric limitations to the method presented in the original paper as follows: A2: LH <= 8*T A3: h = lesser of the inside nozzle projection or 8*(T + te) These limits provide better agreement between the proposed method and FEA for the case of a thick nozzle attached to a very thin shell. The following explanation of the reasoning behind the variable limits of reinforcement in the calculation of A1 may also prove helpful. The equation to use when calculating Lr changes depending on whether or not a reinforcement pad is present and also on the width and thickness of the pad itself. This is based on the FEA observation that if the pad is large enough it behaves locally like a thicker shell. It should be pointed out that the FEA excludes any stress analysis in the nozzle attachment welds. In the author's opinion this means that in order to take advantage of the larger A1 limit (LR = 8*(T+te)), a full penetration pad to nozzle weld should be used. Graphical results of this additional verification follow for 6 inch, 12 inch and 24 inch schedule 80 nozzles.
6" schedule 80 Nozzle (SA-106C), internal pressure selected such that PR/T in shell = 20,000 psi 6
5 à
e v i t a v r e s n o C y l g n i s a e r c n I
4
3
2
1
0 24
32
36
48
48
48
60
64
72
72
80
96
96
96
96 120 128 144 144 192 192 240 288 384
D\t PL\(1.5*Sa) (FEA)
PL\(1.5*Sa) (Proposed)
Ar\Aa (VIII-1)
Ar\Aa (VIII-2)
12" schedule 80 Nozzle (SA-106C), internal pressure selected such that PR/T in shell = 20,000 psi 9 8 7 à
e v i t a v r e s n o C y l g n i s a e r c n I
6 5 4 3 2 1 0 24
32
36
48
48
48
60
64
72
72
80
96
96
96
96 120 128 144 144 192 192 240 288 384
D\t PL\(1.5*Sa) (FEA)
PL\(1.5*Sa) (Proposed)
Ar\Aa (VIII-1)
Ar\Aa (VIII-2)
24" schedule 80 Nozzle (SA-106C), internal pressure selected such that PR/T in shell = 20,000 psi 6
5 à
e v i t a v r e s n o C y l g n i s a e r c n I
4
3
2
1
0 36
48
48
60
64
72
72
80
96
96
96
120
128
144
144
192
192
240
D\t PL\(1.5*Sa) (FEA)
PL\(1.5*Sa) (Proposed)
Ar\Aa (VIII-1)
Ar\Aa (VIII-2)
288
384
